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Vectors_Tensors_07_Vector_Calculus_2_Integration

# Vectors_Tensors_07_Vector_Calculus_2_Integration - Section...

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Section 1.7 Solid Mechanics Part III Kelly 50 1.7 Vector Calculus 2 - Integration 1.7.1 Ordinary Integrals of a Vector A vector can be integrated in the ordinary way to produce another vector, for example () {} 3 2 1 2 1 3 2 2 1 2 3 2 15 6 5 3 2 e e e e e e + = + dt t t t 1.7.2 Line Integrals Discussed here is the notion of a definite integral involving a vector function that generates a scalar. Let 3 3 2 2 1 1 e e e x x x x + + = be a position vector tracing out the curve C between the points 1 p and 2 p . Let f be a vector field. Then + + = = C C p p dx f dx f dx f d d 3 3 2 2 1 1 2 1 x f x f is an example of a line integral. Example (of a Line Integral) A particle moves along a path C from the point ) 0 , 0 , 0 ( to ) 1 , 1 , 1 ( , where C is the straight line joining the points, Fig. 1.7.1. The particle moves in a force field given by ( ) 3 2 3 1 2 3 2 1 2 2 1 20 14 6 3 e e e f x x x x x x + + = What is the work done on the particle? Figure 1.7.1: a particle moving in a force field C f x d

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Section 1.7 Solid Mechanics Part III Kelly 51 Solution The work done is ( ) {} + + = = C C dx x x dx x x dx x x d W 3 2 3 1 2 3 2 1 2 2 1 20 14 6 3 x f The straight line can be written in the parametric form t x t x t x = = = 3 2 1 , ,, s o t h a t () 3 13 6 11 20 1 0 2 3 = + = dt t t t W or 3 13 3 2 1 = + + = = C C dt dt dt d W e e e f x f If C is a closed curve, i.e. a loop, the line integral is often denoted C d x v . Note : in fluid mechanics and aerodynamics, when v is the velocity field, this integral C d x v is called the circulation of v about C 1.7.3 Conservative Fields If for a vector f one can find a scalar φ such that = f ( 1 . 7 . 1 ) then (1) 2 1 p p d x f is independent of the path C joining 1 p and 2 p (2) 0 = C d x f around any closed curve C In such a case, f is called a conservative vector field and is its scalar potential 1 . For example, the work done by a conservative force field f is ) ( ) ( 1 2 2 1 2 1 2 1 2 1 p p d dx x d d p p p p i i p p p p = = = = x x f which clearly depends only on the values at the end-points 1 p and 2 p , and not on the path taken between them. It can be shown that a vector f is conservative if and only if o f = curl { Problem 3}. 1 in general, of course, there does not exist a scalar field such that = f ; this is not surprising since a vector field has three scalar components whereas is determined from just one
Section 1.7 Solid Mechanics Part III Kelly 52 Example (of a Conservative Force Field) The gravitational force field 3 e f mg = is an example of a conservative vector field. Clearly, o f = curl , and the gravitational scalar potential is 3 mgx = φ . Also, [] () () 1 2 1 3 2 3 3 3 ) ( ) ( 2 1 2 1 p p p x p x mg dx mg d mg W p p p p = = = = x e Example (of a Conservative Force Field) Consider the force field 3 2 3 1 2 2 1 1 3 3 2 1 3 ) 2 ( e e e f x x x x x x + + + = Show that it is a conservative force field, find its scalar potential and find the work done in moving a particle in this field from ) 1 , 2 , 1 ( to ) 4 , 1 , 3 (.

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Vectors_Tensors_07_Vector_Calculus_2_Integration - Section...

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